Universal tilings of the plane by 0-1 -matrices
type: Item
instance of: scholarly article
title: Universal tilings of the plane by 0-1 -matrices
zbMATH Open document ID: 0708.05016
DOI: 10.1016/0012-365X(88)90133-1
author: Knut Dehnhardt
author: Heiko Harborth
published in: Discrete Mathematics
publication date: 1989-01-01T00:00:00Z
review text: A tiling of the plane by black and white unit squares of the square lattice is called (a,b)-universal if constructed using horizontal and vertical translations of a fundamental (m,n)-rectangle and if every one of the \(2^{ab}\) possible distinct (a,b)-rectangles of black and white unit squares occurs somewhere in the tiling. If the fundamental (m,n)- rectangle is as small as possible (implying \(mn=2^{ab})\) the tiling is called optimal (a,b)-universal. The authors prove that for all a,b\(\geq 2\) there exists optimal (a,b)-universal tilings of the plane determined by fundamental \((2^{b(a-1)},2^ b)\)-matrices with O-1 entries. Examples of optimal (3,2)- and (3,3)-universal tilings are shown in black and white, using numerously repeated (16,4) resp. (64,8) fundamental rectangles. The patch of optimal (3,3)-universal tiling exhibited reminds one (approximately) of a section of black-white marbled paper.
Mathematics Subject Classification ID: 05B45
Mathematics Subject Classification ID: 05B20
Mathematics Subject Classification ID: 52C15
zbMATH DE Number: 4162896
zbMATH Keywords: universal matrix tilings
zbMATH Keywords: O-1 matrices
zbMATH Keywords: tiling of the plane
MaRDI profile type: Publication
cites work: Q5578815
cites work: Universal tilings and universal (0,1)-matrices
cites work: Q3686626
cites work: Satins and Twills: An Introduction to the Geometry of Fabrics
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description: scientific article; zbMATH DE number 4162896